Period deformations and Raabe's formulas for generalized gamma and sine functions

Deformations of the multiple gamma and sine functions with respect to their periods are studied. To describe such deformations explicitly, a new class of generalized gamma and sine functions are introduced. In particular, we study the deformations from the viewpoint of multiplication formulas and Raabe's integral formulas for thesé gamma and sine functions. This new class of gamma functions contains Milnor's type multiple gamma functions as a special case.